中英文原文：

Well, you're in a real pickle. You see, a few years ago, the king decided your life would be forfeit unless you tripled the gold coins in his treasury. Fortunately for you, a strange little man appeared and magically performed the feat. He placed handfuls of coins in and out of a magical bag, and sang a strange rhyme:

好吧，你現(xiàn)在處于一個(gè)危險(xiǎn)的困境。 就在前幾年，國王決定要取你性命， 除非你能把他金庫里的金幣數(shù)量 翻三倍才能免你一死。 幸運(yùn)的是，一個(gè)奇怪的小老頭出現(xiàn)了， 并神奇地助你完成了這一壯舉。 他一邊將一把金幣 放進(jìn)一個(gè)神奇的袋子后再拿出來， 一邊唱著一個(gè)奇怪的咒語：

“The more gold goes in and more comes out, as sure as I am me. And in again, and out again, and now it’s it times three!”

“就像‘我即是我’一樣肯定， 金幣放進(jìn)越多，出來越多。 再進(jìn)去，再出來，現(xiàn)在翻了三倍！”

Incredibly, that tripled the coins and saved your life. Were you grateful? Yes. Were you desperate? Yes. Did you promise him your first-born child in exchange for his help? Yes.

很不可思議， 那翻了三倍的金幣救了你的性命。 你感激嗎？是的。 你很絕望嗎？是的。 你有答應(yīng)用你第一個(gè)孩子 作為他幫助的回報(bào)嗎？是的。

Fast forward to today. No sooner have you given birth to a beautiful baby boy than the little man shows up to claim his prize. You cry and beg him not to take the baby. Softening, he begins, “If you can guess my name—” “Banach-Tarski?” you say. “It’s on the front of your shirt.”

很不可思議， 那翻了三倍的金幣救了你的性命。 你感激嗎？是的。 你很絕望嗎？是的。 你有答應(yīng)用你第一個(gè)孩子 作為他幫助的回報(bào)嗎？是的。

“What! That won’t do. Aha. My bag,” he explains, “increases the number of gold coins placed inside it in a very special way. If I take any number of coins and place them in, more will come out. And if I place those in the bag again, the total that comes out will be three times whatever I began with.”

“哈？那不算。啊哈，我的袋子。” 他解釋到。 “里面有種很特殊的方法 可以增加放進(jìn)去的金幣的數(shù)量， 我放進(jìn)去任意數(shù)量的金幣， 會(huì)拿出來更多的金幣， 然后我把現(xiàn)在的這些金幣 再放進(jìn)去一遍， 那么再拿出來的總數(shù) 總會(huì)是我第一次放進(jìn)去的三倍?！?/p>

He takes 13 coins and places them in the bag, then removes the contents. “I’ve used the magic once, not twice,” he says. “Tell me how many coins are in my hand and I’ll have mercy.”

他拿了 13 個(gè)金幣放進(jìn)了那個(gè)袋子， 然后把里面的東西拿了出來。 “我現(xiàn)在只用了一次魔法， 不是兩次哦。”他說道。 “如果你能說出我手上 有多少金幣，我就放過你?！?/p>

How many coins is he holding?

他拿著多少金幣呢？

Pause here to figure it out yourself. Answer in 3

暫停一下，你自己來思考下。 倒計(jì)時(shí) 3

Answer in 2

?倒計(jì)時(shí) 2

Answer in 1

?倒計(jì)時(shí) 1

The bag’s magic works just like what in mathematics is called a “function,” and it’s convenient in both cases to use an arrow to denote the transformation. We can write what we know like this.

那個(gè)袋子的魔法原理 和數(shù)學(xué)上的“函數(shù)”一樣， 在這兩種情況下 都可以很方便的用箭頭來表示這個(gè)轉(zhuǎn)換， 我們可以像這樣把我們知道寫下來。

We want to know what goes in this particular blank.

我們想知道這個(gè)空白處該填入什么

Maybe the bag just multiplies the number of coins by some number. In that case, multiplying by that number twice would be the same as multiplying by 3, which means the multiplier would be the square root of 3. That’s not a whole number, though. And we don’t have bits of gold coins coming out of the bag. Something else is going on.

或許這個(gè)袋子就是把金幣的數(shù)量 乘上了某個(gè)數(shù)， 這樣的話，乘這個(gè)數(shù)兩次， 就等于原本的數(shù)量乘以 3 ， 那這個(gè)乘數(shù)就應(yīng)該是根號(hào) 3 。 不過，這就不是一個(gè)整數(shù)了， 而且那個(gè)袋子里出來的金幣 可沒有零碎的， 所以應(yīng)該不是這個(gè)。

Well, if filling in the blank between 13 and 39 is too hard, maybe we can start with something easier. Can we figure out what’ll happen to 1 coin?

好吧，如果在 13 和 39 之間 填空太難的話， 那我們可以換個(gè)簡單一點(diǎn)的， 我們能看出 從 1 枚金幣開始是怎樣的嗎？

If you use the bag on a single coin twice, you end up with triple; that’s three gold pieces.

一枚金幣進(jìn)出這袋子兩次， 你就會(huì)得到三倍的數(shù)量； 也就是 3 枚金幣。

Because the bag always increases the number of gold coins, the blank must be between 1 and 3, so 2. It’s a start. What’s next?

因?yàn)檫@個(gè)袋子總是會(huì)使金幣的數(shù)量增加， 所以空白處必須是 在 1 和 3 之間，那就是2。 這算是個(gè)開始，那之后呢？

Let’s think about a few other possible starting places.

讓我們看看其他幾種可能，

We already know 2 becomes 3 and that lets us fill in the next blank as well. Now we’re getting somewhere!

我們都知道 2 會(huì)變成 3 ， 這可以幫助我們填寫下一個(gè)空白處， 這樣我們就有了新的進(jìn)展。

We just need to extend this out to 13. Remember the other rule, though: when you put more coins in, you get more coins out. That means the numbers in every column must go in increasing order as well. In other words, because 6 coins become 9, it’s not possible for 4 coins to become 10. Nor could 4 become 5, since 3 becomes 6.

我們只需要擴(kuò)寫到 13 ， 但是還要記得另一條規(guī)則， 你拿出來的金幣要比你放進(jìn)去的多， 也就是說每列數(shù)字都必須是遞增的。 換種說法，因?yàn)?6 枚金幣變成 9 枚， 那么 4 枚金幣就不可能變成 10 枚， 而且因?yàn)?3 枚會(huì)變成 6 枚， 那么 4 枚也不可能變成 5 枚，

So 7 and 8 fill those blanks on the right of 4 and 5, which in turn gives the answer for two more blanks.

所以對(duì)應(yīng) 4 和 5 的空白處 應(yīng)該填入 7 和 8 ， 這樣就又可以推出 另外兩個(gè)空白處的數(shù)字。

Knowing that the numbers go in increasing order in every column, the only choices for the remaining blanks are 19, 20, 22, and 23.

現(xiàn)在知道了每列數(shù)字都是遞增的， 那么剩下的空白處只能是 19 、 20 、22 和 23 。

And look! We have our answer! There must be 22 gold coins in his hand.

看吧！我們得到了答案！ 他手上肯定是 22 枚金幣。

“I’ll give you three guesses,” the little man begins to say.

小老頭說：“我可以給你三次機(jī)會(huì)?！?/p>

“22 coins,” you respond.

你回答道：“22個(gè)?！?/p>

“What?! How did you know?”

“哈？！你怎么知道？”

“I enjoy a good riddle,” you say. “Also, it’s on the back of your shirt.”

“我喜歡有意思的謎題”，你答道。 “而且，答案就在你衣服背面?！?/p>